In an open carbonate system the aqueous solution is in chemical equilibrium with the CO2 of the atmosphere.1

In contrast to the closed system, where the total amount of inorganic carbon (DIC) remains constant when pH changes, the amount of DIC in an open system increases with increasing pH.

Henry’s Law

The relationship between dissolved carbon dioxide, CO2(aq), and carbon dioxide in the gas phase, CO2(g), is simple proportionality expressed by

(1a)

Henry’s law:

CO2(aq) = const ∙ CO2(g)

The proportionality factor is Henry’s constant. But before specifying its value be aware of the quantities (and units) on both sites of this equation. Let’s agree about two things: First, instead of CO2(g) we use the partial pressure PCO2 in atm. Second, instead of CO2(aq) we use the composite carbonic acid H2CO3*. This yields2

Altogether, the system is described by four equilibrium reactions and their corresponding equilibrium constants:3

(2a)

CO2(g) ⇔ H2CO3*

log KH

= -1.47

(2b)

H2CO3* ⇔ H+ + HCO3-

log K1

= -6.35

(2c)

HCO3- ⇔ H+ + CO3-2

log K2

= -10.33

(2d)

H2O ⇔ H+ + OH-

log KW

= -14.0

The chemical species are interrelated as follows:

The four equilibrium reactions – expressed by the law of mass action – constitute the backbone of the mathematical description. This is our next step.

Nonlinear System of Equations

The open CO2-H2O system is characterized by 6 species (or unknowns):

CO2(g), H2CO3*, HCO3-, CO3-2, H+ and OH- (or H2O)

Accordingly, we need 6 equations to solve for them:

(3a)

KH

= {H2CO3*} / PCO2

= 10-1.47

(3b)

K1

= {H+} {HCO3-} / {H2CO3*}

= 10-6.35

(3c)

K2

= {H+} {CO3-2} / {HCO3-}

= 10-10.33

(3d)

Kw

= {H+} {OH-}

= 10-14.0

(3e)

CT

= [H2CO3*] + [HCO3-] + [CO3-2]

(mole balance)

(3f)

0

= [H+] – [HCO3-] – 2 [CO3-2] – [OH-]

(charge balance)

The first four equations are mass-action laws taken from 2a to (2d); the last two equations represent the mole and charge balance. Please note the “asymmetry”: The mass-action laws are based on activities (denoted by curly braces) while the mole- and charge-balance equations rely on molar concentrations (denoted by square brackets).

Note 1. Remove 3a, and what you get is the set of equations that describes the closed system (based on five equations only).

Note 2. CT in 3e is the total inorganic carbon, usually abbreviated by DIC.

[More details about the three equilibrium constants (KH, K1, K2), and how they are implemented in the program’s thermodynamic database, are given here.]

Equilibrium Speciation of the Open CO2-H2O System

For a given partial pressure PCO2, the open CO2-H2O system is completely determined by the set of equations (3a) to (3f). Under normal atmospheric conditions (PCO2 = 0.00039 atm, 25), we get the following equilibrium speciation:4

It’s quite instructive to compare the above result with the closed CO2 system:

Open System

Closed System

input

pCO2 = 3.408

DIC = 1 mM

pH

5.61

4.68

CO2

mM

0.0133

0.979

HCO3-

mM

0.0024

0.021

CO3-2

mM

4.7·10-8

4.8·10-8

DIC

mM

0.0157

1.000

pCO2

3.408

1.54

In an open system you enter the pCO2 (or CO2 partial pressure); in a closed system you enter DIC. (You cannot enter both values independently.) However, you can interchange the roles formally by imitating

an “open system in contact with atmosphere” by entering 0.0157 mM DIC in a closed system

a “closed system with 1 mM DIC” by an “open system with pCO2 = 1.54” 5

The concept of open/closed systems becomes especially relevant when the solution is attacked by acids or bases:

in a open system the CO2 (or pCO2 value) remains constant

in a closed system DIC remains constant (and CO2 changes)

Example: Titration Calculation

The diagram below displays the results of a titration calculation (addition of HCl and NaOH to an open CO2 system). Note how DIC grows exponentially for pH > 5.6.

The more alkaline the solution becomes the more CO2 is sucked from the atmosphere (which increases the DIC). That’s just the opposite behavior of the closed CO2 system.

Remarks

More about the open_ and closed systems (and the difference between them) is given here and as PowerPoint. ↩